[Dirk J. Struik] Lectures on Classical Differentia(BookZZ.org)

Lectures onClassical Differential

GeometrySECOND EDITION

Dirk J. StruikMASSACHUSETTS INSTITUTE OF TECHNOLOGY

DOVER PUBLICATIONS, INC.New York

Copyright 1950, 1961 by Dirk J. Struik.All rights reserved under Pan American and International

Copyright Conventions.Published in Canada by General Publishing Company,

Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.Published in the United Kingdom by Constable and Com-

pany, Ltd.This Dover edition, first published in 1988, is an un-

abridged and unaltered republication of the second edition(1961) of the work first published in 1950 by the Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.

Manufactured in the United States of AmericaDover Publications, Inc., 31 East 2nd Street, Mineola, N.Y.

11501

Library of Congress Cataloging-in-Publication Data

Struik, Dirk Jan, 1894-Lectures on classical differential geometry / Dirk J.

Struik. - 2nd ed.p. cm.

Reprint. Originally published: Reading, Mass. : Addi-son-Wesley Pub. Co., 1961.

Bibliography: p.Includes index.

ISBN 0-486-65609-81. Geometry, Differential. I. Title.

QA641.S72 1988516.3'602-dc 19 87-34903

CIP

CONTENTS

PREFACE . . . . . . . . . . . . . . . . . . . . . . . V

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . Vii

CHAPTER 1. CURVES . . . . . . . . . . . . . . . . . . 11-1 Analytic representation . . . . . . . . . . . . . . . 11-2 Arc length, tangent . . . . . . . . . . . . . . . . . 51-3 Osculating plane . . . . . . . . . . . . . . . . . . 101-4 Curvature . . . . . . . . . . . . . . . . . . . . 131-5 Torsion . . . . . . . . . . . . . . . . . . . . . 151-6 Formulas of Frenet . . . . . . . . . . . . . . . . . 181-7 Contact . . . . . . . . . . . . . . . . . . . . . 231-8 Natural equations . . . . . . . . . . . . . . . . . 261-9 Helices . . . . . . . . . . . . . . . . . . . . . 331-10 General solution of the natural equations . . . . . . . . . 361-11 Evolutes and involutes . . . . . . . . . . . . . . . . 391-12 Imaginary curves . . . . . . . . . . . . . . . . . 441-13 Ovals . . . . . . . . . . . . . . . . . . . . . . 471-14 Monge . . . . . . . . . . . . . . . . . . . . . 53

CHAPTER 2. ELEMENTARY THEORY OF SURFACES . . . . . . . 552-1 Analytical representation . . . . . . . . . . . . . . . 552-2 First fundamental form . . . . . . . . . . . . . . . 582-3 Normal, tangent plane . . . . . . . . . . . . . . . . 622-4 Developable surfaces . . . . . . . . . . . . . . . . 662-5 Second fundamental form. Meusnier's theorem . . . . . . . 732-6 Euler's theorem . . . . . . . . . . . . . . . . . . 772-7 Dupin's indicatrix . . . . . . . . . . . . . . . . . 832-8 Some surfaces . . . . . . . . . . . . . . . . . . . 862-9 A geometrical interpretation of asymptotic and curvature lines . . 932-10 Conjugate directions . . . . . . . . . . . . . . . . 962-11 Triply orthogonal systems of surfaces . . . . . . . . . . . 99

CHAPTER 3. THE FUNDAMENTAL EQUATIONS . . . . . . . . . 1053-1 Gauss . . . . . . . . . . . . . . . . . . . . . 1053-2 The equations of Gauss-Weingarten . . . . . . . . . . . 1063-3 The theorem of Gauss and the equations of Codazzi . . . . . . 1103-4 Curvilinear coordinates in space . . . . . . . . . . . . 1153-5 Some applications of the Gauss and the Codazzi equations . . . 1203-6 The fundamental theorem of surface theory . . . . . . . . . 124

iii

1V CONTENTS

CHAPTER 4. GEOMETRY ON A SURFACE. . . . . . . . . . . 127

4-1 Geodesic (tangential) curvature . . . . . . . . . . . . . 1274-2 Geodesics . . . . . . . . . . . . . . . . . . . . 1314-3 Geodesic coordinates . . . . . . . . . . . . . . . . 1364-4 Geodesics as extremals of a variational problem . . . . . . . 1404-5 Surfaces of constant curvature . . . . . . . . . . . . . 1444-6 Rotation surfaces of constant curvature . . . . . . . . . . 1474-7 Non-Euclidean geometry . . . . . . . . . . . . . . . 1504-8 The Gauss-Bonnet theorem . . . . . . . . . . . . . . 153

CHAPTER 5. SOME SPECIAL SUBJECTS . . . . . . . . . . . 1625-1 Envelopes . . . . . . . . . . . . . . . . . . . . 1625-2 Conformal mapping . . . . . . . . . . . . . . . . 1685-3 Isometric and geodesic mapping . . . . . . . . . . . . 1755-4 Minimal surfaces . . . . . . . . . . . . . . . . . . 1825-5 Ruled surfaces . . . . . . . . . . . . . . . . . . 1895-6 Imaginaries in surface theory . . . . . . . . . . . . . 196

SOME PROBLEMS AND PROPOSITIONS . . . . . . . . . . . . 201

APPENDIX: The method of Pfaffians in the theory of curves and surfaces. 205

ANSWERS TO PROBLEMS . . . . . . . . . . . . . . . 217

INDEX . . . . . . . . . . . . . . . . . . . . 226

PREFACE

This book has developed from a one-term course in differential geometrygiven for juniors, seniors, and graduate students at the MassachusettsInstitute of Technology. It presents the fundamental conceptions of thetheory of curves and surfaces and applies them to a number of examples.Some care is given to historical, biographical, and bibliographical material,not only to keep alive the memory of the men to whom we owe the mainstructure of our present elementary differential geometry, but also to allowthe student to go back to the sources, which still contain many preciousideas for further thought.

No author on this subject is without primary obligation to the twostandard treatises of Darboux and Bianchi, who, around the turn of thecentury, collected the result of more than a century of research, themselvesadding greatly to it. Other fundamental works constantly consulted bythe author are Eisenhart's Treatise, Scheffers' Anwendung, and Blaschke'sVorlesungen. Years of teaching from Graustein's Differential Geometryhave also left their imprint on the presentation of the material.

The notation used is the Gibbs form of vector analysis, which after yearsof competition with other notations seems to have won the day, not only inthe country of its inception, but also in many other parts of the world.Those unfamiliar with this notation may be aided by some explanatoryremarks introduced in the text. This notation is amply sufficient for thosemore elementary aspects of differential geometry which form the subjectof this introductory course; those who prefer to study our subject withtensor methods and thus to prepare themselves for more advanced researchwill find all they need in the books which Eisenhart and Hlavaty havedevoted to this aspect of the theory. Some problems in the present bookmay serve as a preparation for this task.

Considerable attention has been paid to the illustrations, which may behelpful in stimulating the student's visual understanding of geometry. Inthe selection of his illustrations the author has occasionally taken his in-spiration from some particularly striking pictures which have appeared inother books, or from mathematical models in the M.I.T. collection.* The

* In particular, the following figures have been wholly or in part suggested byother authors: Fig. 2-33 by Eisenhart, Introduction; Figs. 2-32, 2-33, 3-3, 5-14by Scheffers, Anwendung; Figs. 2-22, 2-23, 5-7 by Hilbert-Cohn Vossen; andFig. 5-9 by Adams (loc. cit., Section 5-3). The models, from L. Brill, Darmstadt,were constructed between 1877 and 1890 at the Universities of Munich (underA. Brill) and Gottingen (under H. A. Schwarz).

V

vi PREFACE

Art Department of Addison-Wesley Publishing Company is responsiblefor the excellent graphical interpretation and technique.

The problems in the text have been selected in such a way that most ofthem are simple enough for class use, at the same time often conveying aninteresting geometrical fact. Some problems have been added at the endwhich are not all elementary, but reference to the literature may here behelpful to students ambitious enough to try those problems.

The author has to thank the publishers and their adviser, Dr. EricReissner, for their encouragement in writing this book and Mrs. Violet Haasfor critical help. He owes much to the constructive criticism of his classin M 442 during the fall-winter term of 1949-50, which is responsible formany an improvement in text and in problems. He also acknowledges withappreciation discussions with Professor Philip Franklin, and the help ofMr. F. J. Navarro.

DIRK J. STRUM

PREFACE TO THE SECOND EDITION

In this second edition some corrections have been made and an appendixhas been added with a sketch of the application of Cartan's method ofPfaffians to curve and surface theory. This sketch is based on a paperpresented to the sixth congress of the Mexican Mathematical Society,held at Merida in September, 1960.

A Spanish translation by L. Bravo Gala (Aguilar, Madrid, 1955) isonly one of the many tokens that the book has been well received. Theauthor likes to use this opportunity to express his appreciation and tothank those who orally or in writing have suggested improvements.

D.J.S.

BIBLIOGRAPHYENGLISH

EISENHART, L. P., A treatise on the differential geometry of curves and surfaces.Ginn & Co., Boston, etc., 1909, xi + 476 pp. (quoted as Differential Geometry).

EISENHART, L. P., An introduction to differential geometry with use of the tensorcalculus. Princeton University Press, Princeton, 1940, 304 pp.

FORSYTH, A. R., Lectures on the differential geometry of curves and surfaces. Uni-versity Press, Cambridge, 1912, 34 + 525 pp.

GRAUSTEIN, W. C., Differential geometry. Macmillan, New York, 1935, xi + 230PP.

KREYSZIG, E., Differential geometry. University of Toronto Press, Toronto,1959, xiv + 352 pp. Author's translation from the German: Differentialgeometry.Akad. Verlagsges, Leipzig, 1957, xi + 421 pp.

LANE, E. P., Metric differential geometry of curves and surfaces. University ofChicago Press, Chicago, 1940, 216 pp.

POGORELOV, A. V., Differential geometry. Noordhoff, Groningen, 1959,ix + 171 pp. Translated from the Russian by L. F. BORON.

WEATHERBURN, C. E., Differential geometry of three dimensions. UniversityPress, Cambridge, I, 1927, xii + 268 pp.; II, 1930, xii + 239 pp.

WILLMORE, T., An introduction to differential geometry. Clarendon Press,Oxford, 1959, 317 pp.

FRENCHDARBOUx, G., Legons sur la Worie genkrale des surfaces. 4 vols., Gauthier-

Villars, Paris (2d ed., 1914), I, 1887, 513 pp.; II, 1889, 522 pp.; III, 1894, 512 pp.;IV, 1896, 548 pp. (quoted as Legons).

FAVARD, J., Cours de geometrie differentielle locale. Gauthier-Villars, Paris,1957, viii + 553 pp.

JULIA, G., Elements de geometrie infcnitksimale. Gauthier-Villars, Paris, 2d ed.,1936, vii + 262 pp.

RAFFY, L., Legons sur les applications geom triques de l'analyse. Gauthier-Villars, Paris, 1897, vi + 251 pp.

GERMANBLASCHKE, W., Vorlesungen fiber Differentialgeometrie I, 3d ed., Springer, Berlin,

1930, x + 311 pp. (quoted as Differentialgeometrie).BLASCHKE, W., Einfuhrung in die Differentialgeometrie. Springer-Verlag, 1950,

vii + 146 pp.vii

Viii BIBLIOGRAPHY

HAACK, W., Differential-geometrie. Wolfenbiittler Verlagsanstalt. Wolfen-biittel-Hannover, 2 vols., 1948, I, 136 pp.; II, 131 pp.

HLAVATY, V., Differentialgeometrie der Kurven and Flachen and Tensorrechnung.tbersetzung von M. Pinl. Noordhoff, Groningen, 1939, xi + 569 pp.

KOMMERELL, V. UND K., Allgemeine Theorie der Raumkurven and Flachen. DeGruyter & Co., Berlin-Leipzig, 3e Aufl., 1921, I, viii + 184 pp.; II, 196 pp.

SCHEFFERS, G., Anwendung der Differential- and Integralrechnung auf Geometrie.De Gruyter, Berlin-Leipzig, 3e Aufl. I, 1923, x + 482 pp.; II, 1922, xi + 582 pp.(quoted as Anwendung).

STRUBECKER, K., Differentialgeometrie. Sammlung Goschen, De Gruyter,Berlin, I, 1955, 150 pp.; II, 1958, 193 pp.; III, 1959, 254 pp.

ITALIANBIANCHI, L., Lezioni di geometria differenziale. Spoerri, Pisa, 1894, viii + 541

pp.; 3e ed., I, 1922, iv + 806 pp.; II, 1923, 833 pp. German translation by M.LuKAT; Vorlesungen caber Differentialgeometrie. Teubner, Leipzig, 2e Aufl., 1910,xviii + 721 pp. (Lezioni I quoted as Lezioni.)

RUSSIAN

FINIKOV, S. P., Kurs differencial'noi geometrii. Moscow, 1952, 343 pp.RASEVSKIr, P. K., Kurs differencial'noi geometrii. Moscow, 4th ed., 1956,

420 pp.VYGODSKII, M. YA., Differential'naya geometriya. Moscow-Leningrad, 1949,

511 pp.

There are also chapters on differential geometry in most textbooks of advancedcalculus, such as:

GOURSAT, E., Cours d'analyse mathematique. Gauthier-Villars, Paris, I, 5e ed.,1943, 674 pp. English translation by E. R. HEDRICK, Course in mathematicalanalysis. Ginn & Co., Boston, I, 1904, viii + 548 pp.

The visual aspect of curve and surface theory is stressed in Chapter IV ofD. HILBERT & S. COHN-VossEN, Anschauliche Geometrie. Springer, Berlin, 1932,viii + 310 pp.

The best collection of bibliographical notes and references in: Encyklopadie derMathematischen Wissenschaften. Teubner, Leipzig, Band III, 3 Teil (1902-'27),606 pp., article by H. V. MANGOLDT, R. V. LILIENTHAL, G. SCHEFFERS, A. Voss,H. LIEBMANN, E. SALKOWSKI.

Also :

PASCAL, E., Repertorium der Hoheren Mathematik. 2e Aufl. Zweiter Band,Teubner, Leipzig. Berlin, I (1910), II (1922), article by H. LIEBMANN andE. SALKOWSKI.

The history of differential geometry can be studied in COOLIDGE, J. L., A historyof geometrical methods. Clarendon Press, Oxford, xviii + 451 pp., especially pp.318-387.

Lectures onClassical Differential

Geometry

CHAPTER 1

CURVES

1-1 Analytic representation. We can think of curves in space as pathsof a point in motion. The rectangular coordinates (x, y, z) of the pointcan then be expressed as functions of a parameter u inside a certain closedinterval:

x=x(u), y=y(u), z=z(u); u1

2 CURVES [CH. 1

x = a cos u, y = a sin u, z = bu; a, b constants. (1-4)

This curve lies on the cylinder x2 + y2 = a2 and winds around it in sucha way that when u increases by 2ir the x and y return to their original value,while z increases by 2irb, the pitch of the helix (French: pas; German:Ganghohe). When b is positive the helix is right-handed (Fig. 1-2a); whenb is negative it is left-handed (Fig. 1-2b). This sense of the helix is inde-pendent of the choice of coordinates or parameters; it is an intrinsic prop-erty of the helix. A left-handed helix can never be superimposed on aright-handed one, as everyone knows who has handled screws or ropes.

The functions x;(u) are not all constants. If two of them are constantsEqs. (1-1) represent a straight line parallel to a coordinate axis. We alsosuppose that in the given interval of u the functions x;(u) are single-valuedand continuous, with a sufficient number of continuous derivatives (firstderivatives in all cases, seldom more than three). It is sufficient for thispurpose to postulate that there exists at a point P of the curve, where

au

i

P/+---------

ILL:

BFic. 1-2(a) FIG. 1-2(b)

1-1] ANALYTIC REPRESENTATION 3

u = uo, a set of finite derivatives x4+1)(uo), n sufficiently large. Then wecan express xi(uo + h) as follows in a Taylor development:

xi(uo + h) = xi(u)= xi(uo) + 1 xi(uo) + h' xi(uo) + ... + ni xjn)(uo) + o(h'), (1-5)

where the zi, zi, . . . xi() represent derivatives with respect to u and o(h*)is a term such that

A

lim o h = 0.") h^

This is always satisfied, for all values of n, n > 0, when the xi(u) arecomplex functions of a complex u and the first derivatives 1i exist. Thefunctions xi(u) are then analytic. However, we usually consider the xi as realfunctions of a real variable u. The curve (1-1) with the conditions (1-5) isbetter called an arc of curve, but we shall continue to use the term curve aslong as no ambiguity occurs. Points where all ii vanish are called singular,otherwise regular with respect to u. When speaking of points, we meanregular points. When we replace the parameter u by another parameter,

u = J (u1), (1-6)

* See e.g. P. Franklin, A treatise on advanced calculus, John Wiley and Sons,New York, 1940, p. 127; Ch. J. de la Vall a Poussin, Cours d'analyse infcnitoWmale,Dover Publications, New York, 1946, Tome I, 8th ed., p. 80.

4 CURVES [cHi. 1

we postulate that f(u1) be differentiable; when du/dul 0 0 regular pointsremain regular.

Curves can also be defined in ways different from (1-1). We can use theequations

F1(x, y, z) = 0, F2(x, y, z) = 0, (1-7)or

y = fi(x), z = f2(x), (1-8)to define a curve. The type (1-8) can be considered as a special form of(1-1), x being taken as parameter. We obtain it from (1-1) by eliminatingu from y and x, and also from z and x. This is always possible whendx/du 0 0, so that u can be expressed in x. Type (1-8) expresses thecurve C as the intersection of two projecting cylinders (Fig. 1-3). As toEqs. (1-7), they define two implicit functions y(x) and z(x) when thefunctional determinant

F1F2 _ aFl aF2 _ aF2 aF1 54 0.*Y z) az ay az ay

This brings us to (1-8) and thus to (1-1).The representations (1-7) and (1-8) define the space curve as the inter-

section of two surfaces. But such an intersection may split into severalcurves. If, for instance, F1 and F2 represent two cones with a commongenerating line, Eqs. (1-7) define this line together with the remainingintersection. And if we eliminate u from the equations

x=u, y=u2, z=u3, (1-9)which represent a space curve C ofthe third degree (a cubic parabola),we obtain the equations

y=x2, xz-y2=0,which represent the intersection of acylinder (Fig. 1-4) and a cone. Thisintersection contains not only thecurve (1-9), but also the Z-axis.

The complete intersection of analgebraic surface of degree m and analgebraic surface of degree n is aspace curve of degree mn, which may

* Franklin, loc. cit., pp. 340-341. FIG. 1-4

1-21 ARC LENGTH, TANGENT 5

split into several curves, the degree of which adds up to mn. We say that a sur-face is of degree k when it is intersected by a line in k points or, what is thesame, by a plane in a curve of degree k. A space curve is of degree 1 if a planeintersects it in l points. The points of intersection may be real, imaginary,coincident, or at infinity. In the case (1-9) we substitute x, y, z into theequation of a plane ax + by + cz + d = 0, and obtain a cubic equation for u,which has three roots, indicating three points of intersection.

This explains why we often prefer to give a curve by equations of theform (1-1). Moreover, this presentation allows a ready application of theideas of vector analysis.

1zl tF hi e1, e2, e3i oror t s purpose, efor short e; (i = 1, 2, 3) be unitvectors in the direction of the positiveX, Y, and Z-axes. Then we can givea curve C by expressing the radiusvector OP = x of a generic point Pas a function of u (Fig. 1-5) in thefollowing way:

x = x1e1 + x2e2 + x3e3, (1-10) e2e1where the x; are given by Eq. (1-1). /

We indicate P not only by P(xi), xbut also by P(x) or P(u); we shallalso speak of "the curve x(u)."

Fia. 1-5

The xi are the coordinates of x, the vectors x1e1, x2e2, x3e3 are the com-ponents of x along the coordinate axes. We shall often indicate a vectorby its coordinates, as x(x, y, z), or as x(xi).

The length of the (real) vector x is indicated byxi = -1141 + 42 + xs (1-11)

1-2 Arc length, tangent. We suppose in this and in the next sections(until Sec. 1-12) that the curve C is real, with real u. Then, as shown inthe texts on calculus,* we can express the arc length of a segment of thecurve between points A (uo) and P(u) by means of the integral

s(u) = fv2 + y2 + du = J V du. (2-1)+b

* See e.g. P. Franklin, A treatise on advanced calculus, John Wiley and Sons,New York, 1940, pp. 284, 294; Ch. J. de la Vallee Poussin, Cours d'analyse infini-tesimale, Dover Publications, New York, 1946, Tome I, 8th ed., p. 272.

6 CURVES

The dot will always indicate differentiation with respect to u:

a = dx/du, i; = dx1/du. (2-2)The square root is positive. The expression g g is the scalar product of awith itself; it is always >0 for real curves. x is assumed to be nowherezero in the given interval (no singular points, see Sec. 1-1).

We define the scalar product of two vectors v(vi) and w(w;) by the formula:V W = W V = V1W1 + V2W2 + V8w3

It can be shown that, V being the anglebetween v and w (Fig. 1-6),

v w = JvJJwJ cos 'p. (2-4)This shows that v w = 0 means that vand w are perpendicular. A unit vectoru satisfies the equations

Jul=1, FIG. 1-6

(2-3)

The arc length s increases with increasing u. The sense of increasingarc length is called the positive sense on the curve; a curve with a sense onit is called an oriented curve. Most of our reasoning in differential geom-etry is with oriented curves; our space has also been oriented by the intro-duction of a right-handed coordinate system. However, our results areoften independent of the orientation.

When we change the parameter on the curve from u to ul the arc lengthretains its form, with ul instead of u. We can express this invarianceunder parameter transformations by replacing Eq. (2-1) with the equation

ds2 = dx2 + dye + dz2 = dx dx, (2-5)which is independent of u.

When we now introduce s as parameter instead of u - which is alwayslegitimate, since ds/du 0 0 - then Eq. (2-5) shows us that

dx dx=

'

1d ds (2-6)The vector dx/ds is therefore a unit vector. It has a simple geometrical

interpretation. The vector Ax joins two points P(x) and Q(x + .x) onthe curve. The vector Ax/As has the same direction as Ax and for As -+ 0

1-21 ARC LENGTH, TANGENT

passes into a tangent vector at P (Fig. 1-7).Since its length is 1 we call the vector

t = dx/ds (2-7)the unit tangent vector to the curve at P. Itssense is that of increasing s. Since

dx dx dsdu ds du' (2-8)

we see that 1i = dx/du is also a tangent vector,though not necessarily a unit vector.

We often express the fact that the tangentis the limiting position of a line through Pand a point Q in the given interval of u, whenQ -> P, by saying that the tangent passes 0through two consecutive points on the curve.This mode of expression seems unsatisfactory,but it has considerable heuristic value and canstill be made quite rigorous.

7

Ax

FIG. 1-7

THEOREM. The ratio of the arc and the chord connecting two points P andQ on a curve approaches unity when Q approaches P.Indeed, when As is the are PQ and c is the chord PQ, then for Q -' P

(Fig. 1-8):

As Aslim =1imc N/(AX)2 + (Ay)2 + (AZ)2

0s/Du= lim (yu)z+ (A/Jfy\z + (Qu\z

s

)x2 + y2 + j2 = 1, (2-9) FIG. 1-8

which proves the theorem. It also proves that the ratio of Au and c isfinite.

The ratios Ox/Ds, Ay/Os, t z/Os (Fig. 1-8) therefore approach, for As -> 0,the cosines of the angles which the oriented tangent at P makes with thepositive X-, Y-, and Z-axes. This means that t can be expressed in theform:

t = e1 cos a, + e2 cos a2 + e3 cos a3, (2-10)

S CURVES

which is in accordance with the identity

cost al + cost a2 + cost a3 = 1. (2-11)A generic point A (X) on the tangent line at P is

determined by the equation (Fig. 1-9) :X = x + vt, v = PA, (2-12)

in coordinates (supposing all dxi P` 0)X1 - X1 _ X2 - X2 _ X3 - x3 (2-13)

COS al COS a2 cos a3or

X1 - x1 _ X2 - x2 _ X3 - xdxl dx2 dx3

ExAMPLES. (1) Circle (Fig. 1-10).

(2-14)

x = a cos u, y = a sin u, z = 0; (2-15)z =-a sin u, y = a cos u;s = au + const, take s = au;

x = a cos (s/a), y = a sin (s/a).Unit tangent vector: t(-sin u, cos u).Equation of tangent line:

X1-acosuX2-asinu- sin u coS u

Fm. 1-10B

FIG. 1-9

FIG. 1-11

[CH. I

1-2] ARC LENGTH, TANGENT

or, writing for (X1X2) again (xy):9

x cos u + y sin u = a.

(2) Circular helix (Fig. 1-11).

x = a cos u, y = a sin u, z = bu; (2-16).t =-a sin u, y= a cos u, z=b, s = ua2+b2=cu

tI--sinu,acosu,b)\\ c c C

The tangent vector makes a constant angle a3 with the Z-axis:

cos a3 = b/c, hence tan a3 = a/b.

If B is the intersection of the tangent at P with the XOY-plane, and P3the projection of P on this plane, then

P3B = PP3 tan a3 = bub = au = are AP3.

The locus of B is therefore the involute of the basic circle of the cylinder(see Sec. 1-11).

(3) A space curve of degree four(Fig. 1-12).

x = all -+- cos u), y = a sin u,z = 2a sin u/2.

For 0 < u < 2ir we obtain thepoints above the XOY-plane; for- 27r < u < 0 those below this plane.The whole curve is described for - 27r< u < 27r. Elimination of u gives

x2 + y2 + 32= 4a2,

(x - a)2 + y2 = a2.The curve is the full intersection of

the sphere with center at 0 and radius2a, and the circular cylinder withradius a and axis in the XOZ-plane FIG. 1-12

10 CURVES [CH. 1

at distance a from OZ. The substitution u = tan u/4 allows us to writethe coordinates of the curve by means of rational functions:

2a(1 - u2)2 4au(1 - u2) 4auX (1-+2)2' l' (1+u2)2 z=1+u2.

Substitution of these expressions into the equation of a plane,ax + by + cz + d = 0,

gives a biquadratic equation, showing that the degree of the curve is fourindeed. The are length

s=aJ 1+cos22du0

is an elliptic integral.

The reader will recognize in Fig. 1-12 the so-called temple of Viviani (1692),well known in the theory of multiple integration, remarkable because both thearea and the volume of the hemisphere x > 0 outside the cylinder is rational in a.

1-3 Osculating plane. The tangent can be defined as the line passingthrough two consecutive points of the curve. We shall now try to find theplane through three consecutive points, which means the limiting positionof a plane passing through three nearby points of the curve when two ofthese points approach the third. For this purpose let us consider a planeX a = p; X generic point of the plane, a 1 plane, p a constant, (3-1)passing through the points P, Q, R on the curve given by X = x(uo),X = x(u1), X = X(U2). Then the function

f (u) = x a - p, x = x(u) (3-2)satisfies the conditions

f (uo) = 0, f(U1) = 0, f (U2) = 0.Hence, according to Rolle's theorem, there exist the relations

f(Vi) = 0, f'(v2) = 0, uo < v1 < u1, u1 < V2 < U2,and

f"(v3) = 0, v1 < V3

1-31 OSCULATING PLANE 11

Eliminating a from Eqs. (3-3) and (3-1) we obtain a linear relationbetween X - x, it, it (all 1 a) :

X = x + XI + it, X, arbitrary constants, (3-4a)in coordinates

X1-x1 X2-x2 X3-x3x1 x2 t3 = 0 (3-4b)xl z2 t3

This is the equation of the plane through three consecutive points of thecurve, to which John Bernoulli has given the pleasant name of osculatingplane (German: Schmiegungsebene). It passes through (at least) three con-secutive points of the curve. It also passes through the tangent line (givenby A = 0 in Eq. (3-4a)). The osculating plane is not determined whenit = 0 or when it is proportional to it.

We can express Eqs. (3-4) in another way by the introduction of the vectorproduct of two vectors v and w:

e1 e2 e3

V1 V2 V3

w1 w2 W3= -w x v. (3-5)

It can be shown that, being the angle of v and w,v x w = Ivl lwl sin (p u, (3-6)

where u is a unit vector perpendicular to v and w in such a way that thesense v -+ w --+ u is the same as that of OX -> 0 Y -+ OZ, hence a right-handedone. We always have v x v = 0. When v x w = 0, then w has the directionof v, or w = Xv; in this case we say that v and w are collinear.

The triple scalar product (or parallelepiped product) of three vectors (v, w, u) is:ul u2 u3

V1 V2 V3

W1 W2 W3(vwu) = (wuv) = (uvw) = etc. (3-7)

It is zero when the three vectors (each supposedly 54 0) are coplanar, that is,can be moved into one plane.

The following formula is also useful:

(a x b) (c x d) = (a c) (b d) - (a d) (b c), (3-8)with the special case

(a x b) (a x b) = (a a) (b b) - (a b)2. (3-9)

12 CURVES [CH. 1

The equation of the osculating plane can also be written (comp. Eq. (3-7)with Eq. (3-4b)): (X-x,z,x)=0. (3-10)

As to the two exceptional cases, if they are valid at all points of thecurve then they both are satisfied for straight lines and only for those:

(a) x=0, x=a, x=ua+b(b) it = Xi, x = cex' = cfl(t), x = cf2(t) + d,

where a, b, c, d are fixed vectors.If it = X i ('ii Fl- 0) at one point of the curve, then we call this point

a point of inflection. The tangent at such a point has three consecutivepoints in common with the curve (see Section 1-7).

Since PQ passes into the tangent at P, andQR into the tangent at Q, we say that theosculating plane contains two consecutive tangentlines. This indicates that we may facilitateour understanding of the nature of the oscu- p,lating planes by taking (Fig. 1-13) the pointsP1, P2, P3, ... on the curve and consideringthe polygonal line P1P2.P3... The sidesP1P2, P2P3, ... are all very short, and repre-sent the tangent lines; the planes P1P2P3,P2P3P4, ... represent the osculating planes.Two consecutive tangent lines P1P2j P2P3 liein the osculating plane P1P2P3; two consecu-tive osculating planes P1P2P3, P2P3P4 intersectin the tangent line P2P3, etc.

P1

FIG. 1-13

ExAMPLES. (1) Plane curve. Since in this case the lines PQ and QR, con-sidered in Eq. (3-1), lie in the plane of the curve, the osculating plane coin-cides with the plane of the curve. This is also clear from Fig. 1-13. Whenthe curve is a straight line the osculating plane is indeterminate and may beany plane through the line.

(2) Circular helix. The equation of the osculating plane isX1- acosu X2- asinu X3- bu

- a sin u acosu b- acosu - a sin u 0

or, writing (x, y, z) for (X1, X2, X3):

= 0, (3-10)

bx sin u - by cos u + az = abu.

1-41 CURVATURE 13

This equation is satisfied by x = X cosu, y = X sin u, z = bu for all values ofX, which (by fixed u) shows that theosculating plane at P contains theline PA parallel to the XOY-planeintersecting the Z-axis. The planethrough PA and the tangent at P isthe osculating plane at P (Fig. 1-14).The locus of the lines PA, indicatedby P1A 1, P2A 2, P3A ...... is a sur-face called the right helicoid (see Sec.2-2, 2-8).

1-4 Curvature. The line in theosculating plane at P perpendicularto the tangent line is called the prin-cipal normal (e.g., the lines AP inFig. 1-14). In its direction we placea unit vector n, the sense of which maybe arbitrarily selected, provided it iscontinuous along the curve. If wenow take the are length as parameter:

FIG. 1-14

x = x(s), t = dx/ds = x', t t = 1, (4-1)where the prime signifies differentiation with respect to s, then we obtainby differentiating t t = 1:

t- t' = 0. (4-2)This shows that the vector t' = dt/ds is perpendicular to t, and since

t = x' = u', t' = ii(u')2 + zu", (4-3)we see that t' lies in the plane of z and it, and hence in the osculating plane.We can therefore introduce a proportionality factor K such that

k = dt/ds = Kn. (4-4)

The vector k = dt/ds, which expresses the rate of change of the tangentwhen we proceed along the curve, is called the curvature vector. The factorK is called the curvature; SKI is the length of the curvature vector. Althoughthe sense of n may be arbitrarily chosen, that of dt/ds is perfectly deter-mined by the curve, independent of its orientation; when s changes sign, t

14 CURVES [Cxi. 1

also changes sign. When n (as is often done) is taken in the sense of dt/ds,then K is always positive, but we shall not adhere to this convention (seebelow, small type).

When we compare the tangent vectors t(u) at P 4'---Atand t + zt(u + h) at Q (Fig. 1-15) by moving tfrom P to Q, then t, At and t + At form an isoscelestriangle with two sides equal to 1, enclosing theangle Op, the angle of contingency. Since

jOtI = 2 sin Dp/2= A(P + terms of higher order in Ago,

we find for i,p->0:

1.1 = Idt/dsl = jkj = Idp/dsl, (4-5) FIG. 1-15which is the usual definition of the curvature in the case of a plane curve.

From Eq. (4-4) follows:K2 = x" . x" (4-6)

We define R as K-1. The absolute value of R is the radius of curvature,which is the radius of the circle passing through three consecutive points of thecurve, the osculating circle. To prove it, we first observe that this circlelies in the osculating plane. Let a circle be determined in this plane asintersection of the plane and the sphere given by

(X- c) (X- c) - r2 = 0 (X generic point of the sphere, c center, r radius).This sphere must pass through the points P, Q, R on the curve given byX = x(so), X = x(s1), X = x(s2); the vector c points from 0 to a point inthe osculating plane so that x - c lies in the osculating plane. Reasoningas in Sec. 1-3 on the function

f (s) = (x - c) (x - c) - r2, x = x(s), c, r constants,we find, for the limiting values of c and r, the conditions

As) = 0,f(s) = 0, or (x - c) x' = 0, (4-7)

f"(s) = 0, or (x - c) x"+c lies in the osculating plane, it is a linear combination of x' and

x". Hence

t+ At

x-C=Xx'+x",

1-5] TORSION 15

where X and are determined by Eq. (4-7). We find X = 0, -1 = x" x",so that

c=x "- x

or, in consequence of Eqs. (4-4) and (4-6) :

C = x + Kn/K2 = x + Rn. (4-8)

This shows that the center of the osculating circle lies on the principalnormal at distance RI from P. Though R = K 1 may be positive or nega-tive, the vector Rn is independent of the sense of n, having the sense of thecurvature vector. Its end point is also called the center of curvature.

Eq. (4-6) shows algebraically that the equation of the curve determines K2but not K uniquely. So long as we consider only one radius of curvature, it doesnot make much difference what sign we attach to K. The simplest way is to

take K > 0, that is, to take the sense of the curvatureKn _vector as the sense of n. But the sign of K is of someimportance when we consider a family of curvaturevectors. For instance, if we take a plane curve (withcontinuous derivatives) with a point of inflection at A(Fig. 1-16), then the curvature vectors are pointed indifferent directions on both sides of A and it may beconvenient to distinguish the concave and the convexsides of the curve by different signs of K. The field ofunit vectors n along the curve is then continuous. We

Kn shall meet another example in the theory of surfacesFIG. 1-16 where we have many curvature vectors at one point

(Sec. 2-6).When the curve is plane we can remove the ambiguity in the sign of K by

postulating that the sense of rotation from t to n is the same as that from OX toOY. Then K can be defined by the equation K = dsp/ds, where rp is the angle ofthe tangent vector with the positive X-axis.

1-5 Torsion. The curvature measures the rate of change of the tangentwhen moving along the curve. We shall now introduce a quantity measur-ing the rate of change of the osculating plane. For this purpose we intro-duce the normal at P to the osculating plane, the binormal. In it we placethe unit binormal vector b in such a way that the sense t -' n -p b isthe same as that of OX - OY -- OZ; in other words, since t, n, b aremutually perpendicular unit vectors, we define the vector b by the formula :

b=txn. (5-1)

CURVES [CH. 1

These three vectors t, n, b can be taken as a new frame of reference.16

They satisfy the relations

(5-2)

This frame of reference, moving along the curve, forms the moving trihedron.The rate of change of the osculating plane is expressed by the vector

b' = db/ds.

This vector lies in the direction of the principal normal, since, according tothe equation b t = 0,

b b = 1,0,

so that, introducing a proportionality factor r,db/ds = - rn. (5-3)

We call r the torsion of the curve. It may be positive or negative, likethe curvature, but where the equation of the curve defines only K2, it doesdefine r uniquely. This can be shown by expressing r as follows:

(t x n')= - K1x if. (x, x (K 'x ),)=

or

(x'x"x"')r = x x , x' = dx/ds. (5-4)This formula expresses r in x(s) and its derivatives independently of the

orientation of the curve, since change of s into -s does not affect the right-hand member of (5-4). The sign of r has therefore a meaning for thenonoriented curve. We shall discuss this further below.

The equations

x'(s) = dx/ds = (dx/du)(du/ds) = au' = x(x a)-1/2,x" = %(u')2 + xu" = [(x x)x - (x . x)]( . x)-2,x", = %(u')8 + 3%u'u" + ku'"

,

allow us to express K2 and r in terms of an arbitrary parameter. We findthe formulas

1-5] TORSION

K2 =(x x) 3(xxx)

17

g = dx/du (5-5a)

(5-5b)

We see that K and T have the dimension L-1. Where IK 1I = SRI iscalled the radius of curvature, IT 'I = ITI is called the radius of torsion.However, this quantity ITI does not admit of such a ready and elegantgeometrical interpretation as IRI.EXAMPLES. (1) Circular helix. From Eq. (2-16) we derive, if a2+b2=c,

(- a aX,c sin u, c cos u,

c,

a" - c cos u, - sin u, 0 I, (s parameter, u = C)X... a

c3sin u, - a

c3cos u, 0 .

Hence0 K=a/c2,

(x'x"x`__ bx" x" e

a a--cosu --sinu c'__b

a2C2a . aWsin u - Wcosu

Hence T is positive when b is positive, which is the case when (see Sec. 1-1)the helix is right-handed; r is negative for a left-handed helix. We also seethat K and T are both constants, and from the equations

F K Ta

K2 + T2, b K2 + T

we can derive one and only one circular helix with given K, r and with givenposition with respect to the coordinate axes (change of a into -a does notchange the helix; it only changes u into u +7r).

(2) Plane curve. Since b is constant, T = 0. If, conversely, r = 0,0, or z + Xii + z = 0; A, A functions of s. This is a linear

homogeneous equation in g, which is solved by an expression of the form

x = clfl(S) + c2f2(s),** See e.g. P. Franklin, Methods of advanced calculus, New York: McGraw-Hill

Book Co., 1941, p. 351. A vector equation is equivalent to three scalar equations,so that the result reached for scalar differential equations can immediately betranslated into vector language.

18

hence

CURVES ICH. 1

x = c1F1(s) + c2F2(s) + c3,where the c,, i = 1, 2, 3, are constant vectors and the F;, j = 1, 2, functionsdetermined by the arbitrary X and A. This shows that the curve x(s)lies in the plane through the end point of c3 parallel to cl and C2. Thismeans that x(s) can be any plane curve. For straight lines the torsion isindeterminate.

Curvature and torsion are also known as first and second curvature, andspace curves are also known as curves of double curvature.

The name torsion is due to L. I. Vallt e, Traite de geometrie descriptive, p. 295of the 1825 edition. The older term was flexion. The name binormal is due toB. de Saint Venant, Journal Ecole Polytechnique 18, 1845, p. 17.

1-6 Formulas of Frenet. We have found that t' = Kn and b' = - rn.Let us complete this information by also expressing n' = do/ds in termsof the unit vectors of the moving trihedron. Since n' is perpendicular to n,n n' = 0, and we can express n' linearly in terms of t and b:

n' = alt + alb.Since according to Eq. (5-2)

al =and

we find for do/ds:

The three vector formulas,

dt

ds

b' = r,

n' _ - Kt + rb.

Kn

+ rb (6-1)

- rn

together with dx/ds = t, describe the motion of the moving trihedron alongthe curve. They take a central position in the theory of space curves andare known as the formulas of Frenet, or of Serret-Frenet.

1-6] FORMULAS OF FRENET 19

They were obtained in the Toulouse dissertation of F. Frenet, 1847, of whichan abstract appeared as "Sur les courbes h double courbure," Journal de Mat hem.17 (1852), pp. 437-447. The paper of J. A. Serret appeared Journal de Mathkm.16 (1851), pp. 193-207; it appeared after Frenet's thesis, but before Frenet madehis results more widely known.

The coordinates of t, n, and b are the cosines of the angles which theoriented tangent, principal normal, and binormal make with the positivecoordinate axes. When we indicate this by t(cos a$), n(cos F+i), b(cos tii),i = 1, 2, 3, the Frenet formulas take the following coordinate form:

d3COS ai = K COS $j, ds COS Ni =-K Cos ai + r Cos yi,

s cos yi = - T COS Pi. (6-1a)The three planes formed by the three sides of the moving trihedron

(Fig. 1-17) are called:the osculating plane, through tangentand principal normal, with equation

(y-the normal plane, through principalnormal and binormal, with equation

(y-x)-t=0,the rectifying plane, through binormaland tangent, with equation

(y -

Rectifyingplane

NormalLP--2lane

- -1COsculating plane

FIG. 1-17

If we take the moving trihedron at P as the trihedron of a set of new Car-tesian coordinates x, y, z, then the behavior of the curve near P is expressedby the formulas (6-la) in the form (x" = Kn, x"' = - K2t + K'n + Krb):

x'1, y'=0, z'0,x" = 0, y" = K, z" = 0, (6-2)

X111 =-K2 y111 = K, Z... = KT.

From these equations we deduce for s --> 0

lim y2 = lim 22xx'2x = 2' (6-3)

z z' z" z"' KTlim z = lim 322x' = lim 6x(x,)2 = lim 6(x')s = 6'

20 CURVES [CH. 1

hence

lim z2 T2 8 2 21C 1 = 2 RT2.y3

= -- = 7-2K-136 K3 9 9

b-z

N t=xFIG. 1-19

This shows that the projections of the curve on the three planes of themoving trihedron behave near P like the curves

y = 2 x2 (projection on the osculating plane),z = 6 x3 (projection on the rectifying plane), (6-4)

z2 =

9

r2Ry3 (projection on the normal plane).

Fig. 1-18 shows this behavior in an orthographic projection, takingK > 0, T > 0. If the sign of -r is changed, the projection on the rectifyingplane changes to that of Fig. 1-19. This again shows the geometricalmeaning of the sign of T.

Fig. 1-20 gives a representation of the curve and its trihedron in space.

EXERCISES

1. Find the curvature and the torsion of the curves:(a) x u, y=v2, z=u3.

U1.(b) x=u, y= uu z=1u (Why isr=0?)

1-6] FORMULAS OF FRENET 21

Fla. 1-20

(c) y = f(x), z = 9(x)(d) x = a(u - sin u), y = all - cos u), z = bu.(e) x = a(3u - u3), y = 3aus, z = a(3u + u3) (Here K' = r2.)2. (a) When all the tangent lines of a curve pass through a fixed point, the

curve is a straight line. (b) When they are parallel to a given line the curve is alsoa straight line.

3. (a) When all the osculating planes of a curve pass through a fixed point,the curve is plane. (b) When they are parallel to a given plane the curve is alsoplane.

4. The binormal of a circular helix makes a constant angle with the axis of thecylinder on which the helix lies.

5. Show that the tangents to a space curve and to the locus C of its centers ofcurvature at corresponding points are normal.

6. The locus C of the centers of curvature of a circular helix is a coaxial helixof equal pitch.

7. Show that the locus of the centers of curvature of the locus C of problem 6 isthe original circular helix and that the product of the torsions at correspondingpoints of C and the helix is equal to K', the square of the curvature of the helix.

22 CURVES [CH. I8. All osculating planes to a circular helix passing through a given point not

lying on the helix have their points of contact with the helix in the same plane.9. Prove that the curve

x = a sine u, y = a sin U COs u, z=acosu

lies on a sphere, and verify that all normal planes pass through the origin. Showthat this curve is of degree four.

10. When x = x(t) is the path of a moving point as a function of time, showthat the acceleration vector lies in the osculating plane.

11. Determine the condition that the osculating circle passes through fourconsecutive points of the curve.

12. Show that for a plane curve, for which x = x(s), y = y(s), and the sign of Kis determined by the assumption of p. 15, K = x'y" - x"y'.

13. Starting with the Eq. (1-5) for the curve a(s), derive expansions for the pro-jections on the three planes of the moving trihedron, and compare with Eq. (6-4).

14. Determine the form of the function o(u) such that the principal normals ofthe curve x = a cos u, y = a sin u, z = p(u) are parallel to the XOY-plane.

15. (a) The binormal at a point P of the curve is the limiting position of thecommon perpendicular to the tangents at P and a neighboring point Q, when Q P.Also find (b) the limiting position for the common perpendicular to the binormals.

16. Find the unit tangent vector of the curve given by

F,(x, y, z) = 0, F2(x, y, z) = 0-

17. Transformation of Combescure. Two space curves are said to be obtainablefrom each other by such a transformation if there exists a one-to-one correspondencebetween their points so that the osculating planes at corresponding points areparallel. Show that the tangents, principal normals, and binormals are parallel.(Following L. Bianchi, Lezioni I, p. 50, we call such transformations after E. J. C.Combescure, Annales Ecole Normale 4, 1867, though the transformations discussedin this paper are more specifically qualified, and deal with certain triply orthogonalsystems of curves.)

18. Cinematical interpretation of Frenet's formulas. When a rigid body rotatesabout a point there exists an axis of instantaneous rotation, that is, the locus of thepoints which stay in place. Show that this axis for the moving trihedron (we donot consider the translation expressed by dx = t ds) has the direction of the vectorR = rt + Kb, so that the Frenet formulas can be written in the form

t'=Rxt, n'=Rxn, b'=Rxb.This constitutes the approach to the theory of curves (and surfaces) typical ofG. Darboux and E. Cartan, the "m6thode du tribdre mobile." (Compare G. Dar-boux, Legons I; E. Cartan, La methode du repere mobile, Actualitess scientifiques194, Paris, 1935.)

1-71 CONTACT 23

19. Spherical image. When we move all unit tangent vectors t of a curve C to apoint, their end points will describe a curve on the unit sphere, called the sphericalimage (spherical indicatrix) of C. Show that the absolute value of the curvature isequal to the ratio of the arc length dst of the spherical image and the arc length ofthe curve ds. What is the spherical image (a) of a straight line; (b) of a planecurve; (c) of a circular helix?

20. Third curvature. When we extend the operation of Exercise 19 to the vectorsn and b, we obtain the spherical image of the principal normals and of the binormals.If ds and dsb represent the elements of are of these two images respectively, showthat =+1/K2 + r2 and Idsb/dsj = ITI. The quantity K2 V+ r2 is some-times called the third (or total) curvature.

1-7 Contact. Instead of stating that figures have a certain number ofconsecutive points (or other elements) in common, we can also state thatthey have a contact of a certain order. The general definition is as follows(Fig. 1-21):

Let two curves or surfaces 2;1, 2;2 have a regular point P in common. Takea point A on 2;1 near P and let AD be its distance to 12 Then 12 has a con-tact of order n with F+1 at P, when for A -' P along E1

lim (Ap k

is finite (V-0) for k = n + 1, but =0for k = n. [AD = o((AP)k) for k = 1,2, .., nJ

When M1 is a curve x(u) and 12 asurface (Fr, F,,, Fs not all zero)

FIG. 1-21

E1

F(x, y, z) = 0, (7-1)we make use of the fact that the distance AD of a point A (x1, y1, z1) near Pis of the same order as F(xi, yi, z1). The general proof of this fact requiressome surface theory, but in the case of the plane and the sphere, the onlycases we discuss in the text, it can be readily demonstrated (see Exercise 4,Sec. 1-11).

Let us now consider the function obtained by substituting the x; of thecurve E1 into Eq. (7-1):

f(u) = F[x(u), y(u), z(u)]. (7-2)This procedure is simply a generalization of the method used in Sees. 1-3

and 1-4 to obtain the equations of the osculating plane and the osculatingcircle. Let f (u) near P(u = uo) have finite derivatives f (') (uo), i = 1, 2,

24 CURVES [CH. 1

..., n + 1. Then if we take u = u, at A and write h = ul - uo, thenthere exists a Taylor development of f(u) of the form (compare Eq. (1-5)):

2 hn+lf(ul) = f(uo) + hf'(uo) + h2)f (uo) + . + (n + 1)i P+1) (uo) + o(hn+1).

(7-3)Here f(uo) = 0, since P lies on 2;2, and h is of order AP (see theorem

Sec. 1-2) ; f (u,) is of the order of AD. Hence necessary and sufficient condi-tions that the surface has a contact of order n at P with the curve are that at Pthe relations hold:

f(u) = f'(u) = f"(u) = . . . = f(n)(u) = 0; f(n+l)(u) T 0. (7-4)In these formulas we have replaced uo by u.

In the same way we find, if E2 is a curve defined by

Fi(x, y, z) = 0, F2(x, y, z) = 0,that necessary and sufficient conditions for a contact of order n at P betweenthe curves are that at P

AM = fi(u) = ... = fi")(u) = 0, (7-5)f2(u) = f2(u) = ... = fr)(u) = 0,

where

fl(u) = FI[x(u), y(u), z(u)], f2(u) = F2[x(u), y(u), z(u)]and at least one of the two derivatives fJn+I)(u), fP+I)(u) at P does not vanish.We can develop similar conditions for the contact of two surfaces.

Instead of AD we can use segments of the same order, making, for in-stance, PD = PA (then L D no longer 90).

If we compare these conditions with our derivation of osculating planeand osculating circle, then we see that they are, in these cases, identicalwith the condition that E, and E2 have n + 1 consecutive points in common.And so we can say that, in general:

Two figures M, and 12, having at P a contact of order n, have n + 1 consecu-tive points in common.

Indeed, following again a reasoning analogous to that of Secs. 1-3 and1-4, and confining ourselves to the case expressed by Eq. (7-1), let usdefine F with n + 1 independent parameters. These are just enough to letsurface E2 pass through (n + 1) points (uo, ul, . . ., un). If these n + 1points come together in point u = uo, then the n + 1 equations (7-4) aresatisfied; if there were more such equations (7-4), then we could determinethe parameters in F so that F+2 would pass through more than n + 1 points.

1-7] CONTACT 25

A similar reasoning holds for the other cases of contact. From this andthe theorems of Sees. 1-3 and 1-4 follows:

A tangent has a contact of (at least) order one with the curve.An osculating plane and an osculating circle have a contact of (at least)

order two with the curve.

The study of the contact of curves and surfaces was undertaken in considerabledetail in Lagrange's Traite des fonctions analytiques (1797) and in Cauchy'sLesons sur les applications du calcul infinitesimal a la geometric I (1826).We shall now apply this theory to find a sphere passing through four

consecutive points of the curve, the osculating sphere. Let this sphere begiven by the equation

(X- c) (X-c)-r2=0, (X generic point of the sphere, c its center, r radius).Consider, in accordance with Eq. (7-2) :

f(s) = (x - c) (x - c) - r2, x = x(s).Then the Eqs. (7-4) take the form, apart from f(s) = 0:

f(s) = 0, or (x -f"(s) = 0, or (x - c) Kn + 1 = 0,f"'(s) = 0, or (x - C) (K'n - K2t + Krb) = 0,

or (r 516 0)(x - (x - (x -The center 0 of the osculating sphere is thus uniquely determined by

c = x + Rn + TR'b. (7-6)This sphere has a contact of order three with the curve. Its intersection

with the osculating plane is the osculating circle. Its center lies in thenormal plane (Fig. 1-22) on a line parallel tothe binormal, the polar axis. The radius of bthe osculating sphere is

Or = R2 ++ (TR')2. (7-7) R,

When the curve is of constant curvature (not-R

a circle), the center of the osculating sphere co- P O' nincides with the center of the osculating circle.

The result expressed by Eq. (7-6) is due toMonge (1807), see his Applications (1850), p. 412. tMonge's notation is quite different. FIG. 1-22

26 CURVES [CH. 1

1-8 Natural equations. When a curve is defined by an equationx = x(s), its form depends on the choice of the coordinate system. When acurve is moved without change in its shape, its equation with respect to thecoordinate system changes. It is not always immediately obvious whethertwo equations represent the same curve except for its position with respectto the coordinate system. Even in the simple case of equations of thesecond degree in the plane (conics) such a determination requires somework. The question therefore arises: Is it possible to characterize a curveby a relation independent of the coordinates? This can actually be ac-complished; such an equation is called natural or intrinsic.

It is easily seen that a relation between curvature and arc length gives anatural equation for a plane curve. Indeed, if we give an equation of theform

K = K(S),

then we find, using the relations

R-1 = K = dcp/ds, cos p = dx/ds,

that x and y can be found by two quadratures:

(8-1)

sin ' = dy/ds,

N V 8

x = R cos s dcp, y = R sin (p djp, lp = K ds. (8-2)wu Wo so

Change of integration constant in x and y means translation, change ofintegration constant in p means rotation of the curve, and thus we canobtain all possible equations in rectangular coordinates, selecting in eachcase the most convenient one for our purpose.

This representation of a curve by means of K (or R) and s goes back to Euler,who used it for special curves: Comment. Acad. Petropolit. 8, 1736, pp. 66-85.The choice of K and s as natural coordinates can be criticized, since s still con-tains an arbitrary constant and K is determined but for the sign. G. Scheffershas therefore developed a system of natural equations of a plane curve in whichd(K2)/drp is expressed as a function of K2. See Anwendung 1, pp. 84-91.

EXAMPLES. (1) Circle: K = a = const.x=Rsinp, y= -Rcossp, ifu=gyp-2,x = R cos u, y = R sin u. (Fig. 1-23)

When a = 0 we obtain a straight line.(2) Logarithmic spiral.

R = as + b = s cot a + p csc a (a, p constants),

1-8] NATURAL EQUATIONS 27

FIG. 1-24

FIG. 1-23

s cosa + p, (selecting s = 0 for ' = a),p = a -1- (tan a In

R = p(csc a) exp (0 cot a), 0 = - a. (Fig. 1-24)Introducing polar coordinates x + iy = re', we find

r = p exp (B cot a) (x = p, y = 0 for s = 0, rp = a).(3) Circle involute:

R2 = gas; rp = 2 a N/s, s =R = arp, x = a cos w d(p = a(cos rp + p sin (p),

y= aJ rp sin (p dip = a(sin p - p cos gyp). (Fig. 1-25)

(4) Epicycloid. We start with the equations in x and y:

x = (a + b) cos - b cos b bby = bsin a b+b

s=4b(a+b)',a

cos-,2bR=

4b(a + b) sin a'a + 2b 2b

28 CURVES [CH. 1

FIG. 1-25

FIG. 1-26

FIG. 1-27

Hence the natural equation is:

zs _ _

A2+ B2 - 1'where

(8-3)

A=

4b(a + b), B = 4b(a + b). A > B. (Fig. 1-26) (8-4)a ' a + 2b '

When A < B we obtain a hypocycloid. An epicycloid is the locus of apoint on the circumference of a circle when rolling on a fixed circle on theoutside; a hypocycloid when rolling on the inside. When A/B is rational,the curve is closed. When a - oo, A -, 4b, B -- 4b, and we obtain

s2 + R2 = 16b2,

the equation of a cycloid, obtained by rolling a circle of radius b on astraight line (Fig. 1-27). We now prove the

1-81 NATURAL EQUATIONS 29

FUNDAMENTAL THEOREM for space curves:If two single-valued continuous functions K(s) and r(s), s > 0, are given,

then there exists one and only one space curve, determined but for its positionin space, for which s is the arc length (measured from an appropriate point onthe curve), K the curvature, and r the torsion.

The equations K = K(s), r = r(s) are the natural or intrinsic equations ofthe space curve.

The proof is simple when we confine ourselves to analytic functions.Then we can write, in the neighborhood of a point s = so, h = s - so:

x(s) = x(so) + x'(so) + h2 X"(8-) + ...

provided the series is convergent in a certain interval s1 < so < s2. Then,substituting for x', x", etc., their values with respect to the moving tri-hedron at P(so), we obtain

x' = t, x" = Kn, x,,, = -K2t + K'n + Krb,

so that

x,,,,

x(s) = x(so) + ht + jKh2n + jh(- K2t + K 'n + Krb) + , (8-5)where all terms can be successively found by differentiating the Frenetformulas, and all successive derivatives of K and r taken, as well as t, n, byat P(so) are supposed to exist because of the analytical character of thefunctions. If we now choose at an arbitrary point x(so) an arbitrary setof three mutually perpendicular unit vectors and select them as t, n, bythen Eq. (8-5) determines the curve uniquely (inside the interval of con-vergence).

It is, however, possible to prove the theorem under the sole assumptionthat K(s) and r(s) are continuous. In this case we apply to the system ofthree simultaneous differential equations of the first order in y,

da= K0i d = - ica + 7-f, L = - r#, (8--6)TS ds ds

the theorem concerning the existence of solutions.

This theorem is as follows. Given a system of differential equations

y,), i= 1,2,....n,dx

where the f; are single-valued and continuous in their n + 1 arguments inside agiven interval (with a Lipschitz condition, satisfied in our case). Then there

30 CURVES [CH. 1

exists a unique set of continuous solutions of this system which assumes givenvalues y, yz..... y when x = x0.*

We deduce from this theorem that we can find in one and in only one waythree continuous solutions ai(s), 01(s), yi(s) which assume for s = so thevalues 1, 0, 0 respectively. We can similarly find three continuous solu-tions a2, 02, 72, so that

a2(so) = 0, l32(so) = 1, 'Y2(50) = 0,and three more continuous solutions a3, 03, 73, so that

a3(80) = 0, 03(60) = 0, 73(50) = 1.

The Eqs. (8-6) lead to the following relations between the a, 0,,y:

2 d (a2 + F+1 + 71) = Kola, - ICalo1 + r7'iF3l - TYlyl = 0,or

+ 01 + 71 = const = 1 + 0 + 0 = 1. (8-7a)Similarly, we find two more relations of the same form:

a2 + 02 + 72 = 1, a32 2+ 03 + 73 = 1, (8-7b)and the three additional relations:

ala2 + 0q102 + 7172 = Ofa1a3 + 13103 + 7173 = 0,a2a3 + 92/33 + 7273 = 0. (8-8)

We have thus found a set of mutually perpendicular unit vectorst(ala2a3), n(/31$2$3), b(7172y3),

where the ai, 13i, yi all are functions of the parameter s (i = 1, 2, 3).This is the consequence of the theorem that if the relations (8-7) and (8-8)

hold, the relationsEai = 2;#i = Zy, = 1, Mai3i = Lai7i = ZQi7i = 0

also hold. This means geometrically that, when t, n, b are three mutuallyorthogonal unit vectors defined with reference to the set ei, e2, e3, then el, e2, e3are three mutually orthogonal unit vectors defined with reference to the sett, n, b.

* Compare E. L. Ince, Ordinary differential equations, Longmans, Green and Co.,London, 1927, p. 71.

1-8] NATURAL EQUATIONS 31

There are oo I trihedrons (t, b, n). If we now integrate t, then theequation

x =ft ds (8-9)determines a curve which has not only t as unit tangent vector field, butbecause of Eq. (8-6) also has (t, n, b) as its moving trihedron, K and T beingits curvature and torsion, and s, because of Eq. (8-9), its arc length.Hence there exists one curve C with given K(s) and T(s) of which the movingtrihedron at P(so) coincides with the coordinate axes.

We now must show that every other curve C which can be brought into aone-to-one correspondence with C such that at corresponding points, givenby equal s, the curvature and torsion are equal, is congruent to C. Thismeans that C can be made to coincide with C by a motion in space. Let usmove the point s = 0 of C to the point s = 0 of C (the origin) in such a waythat the trihedron (t, n, b) of C coincides with the trihedron (t, n, b) of C(the system e1, e2, e3). Let (x;, ;, i3,, Ti) and (x;, a;, j3 , -y;) now denote thecorresponding elements of the moving trihedron of C and C respectively.Then Eq. (8-6) holds for (;, 4j, y;) and for (ai, 13i, -y;) with the same K(s)and T(s). Hence (we omit the index i for a moment):

da+ada+jdO-+0do+Y +y = K$+Ka/3- K/3a + 15yds ds ds ds ds ds

/ /- KN + T - T--yFiq - TyFi = 0,

or

a + 0n + yy = const.

This constant is 1, since it is 1 for s = 0. For the a;, ... , -y; the equationshold:

aid; + 0th. + y:y. = 1, .,2 + 0,1 + y1 = 1, ai + + Y, = 1,which is equivalent to the statement that the three vector pairs (a;, /3j, y;),(;, 0;, -y;) make the angle zero with one another. Hence a; = ;, O; = Vii;,yc = Ti for all values of s, so that

dds (x; - x;) = 0.

This shows that x; - x; = const, but this constant is zero, since it is zerofor s = 0. The curves C and C coincide, so that the proof of the funda-mental theorem is completed.

32 CURVES [CH. 1

All curves of given K(s) and r(s) can thus be obtained from each other bya motion of space. The resulting curves are at least three times differ-entiable.

EXAMPLES. (1) Plane curve. Here K(S) may be any function, r = 0. Inparticular, if K is constant, we find from Eq. (8-2) the circle.

(2) Circular helix. K = const, r = const. We see this immediatelyfrom Example 1, Sec. (1-5), since

a = K2+

r2 and b = K2 + r2

uniquely determine the curvex = a cos s/c, y = a sin s/c, z = bs/c, c = a2 + b2,

and all other curves of the same given K and r must be congruent to thiscurve.

Another way to show this is indicated in Sec. 1-9.(3) Spherical curves (curves lying on a sphere). These are all curves

which satisfy the differential equation in natural coordinates

R2 + (TR')2 = a2, a = any constant. (8-10)Indeed, when a curve is spherical, its osculating spheres all coincide withthe sphere on which the curve lies, hence Eq. (8-10) holds, where a is theradius of the sphere (according to Eq. (7-7)). Conversely, if Eq. (8-10)holds, then the radius of the osculating sphere is constant. Moreover,according to Eq. (7-6):c' = t + (- t + R rb) + R'n + (TR')'b - R'n = {Rr+(TR')'}b = 0. (8-11)Differentiation of Eq. (8-10) shows that for -r5-6 0, R' o 0,

Rr + (TR')' = 0, (8-12)so that c' = 0, and this means that the center of the osculating sphere re-mains in place (except for r = 0, R' = 0, the circle). Eq. (8-12) is thereforethe differential equation of all spherical curves. The circle fits in for

= 0, R' = 0, provided TR' = 0.From Eq. (8-11) it follows, incidentally, that for a nonspherical curve

(not plane) the tangent to the locus of the centers of the osculating sphereshas the direction of the binormal.

For more information on natural equations see E. Cesaro, Lezioni di geometriaintrinseca, Napoli, 1896; translated by G. Kowalewski under the title: Vorlesun-gen caber natiirliche Geometrie, Leipzig, 1901, 341 pp. See also L. Braude, Lescoordonnees intrinseques, collection "Scientia," Paris, 1914, 100 pp. Other

1-91 HELICES 33

forms of the natural equations of a space curve can be found in G. Scheffers,Anwendung I, pp. 278-287, where the fundamental theorem is proved for(dK/ds)2 = f(K2), r = f(K2).

1-9 Helices. The circular helix is a special case of a larger class ofcurves called (cylindrical) helices or curves of constant slope (German:Boschungslinien). They are defined by the property that the tangent makesa constant angle a with a fixed line 1 in space (the axis). Let a unit vectora be placed in the direction of 1 (Fig. 1-28). Then a helix is defined by

t a = cos a = const.

Hence, using the Frenet formulas:

a is therefore parallel to the rectifying plane of the curve, and can be writtenin the form (Fig. 1-29):

a=tcosa+bsina,which, differentiated, gives

0 = Kn cos a - rn sin a = (K cos a - r sin a)n,or

K/r = tan a, constant.For curves of constant slope the ratio of curvature to torsion is constant.

Conversely, if for a regular curve this condition is satisfied, then we canalways find a constant angle a such that

n(K cos a - r sin a) = 0,ds (t cos a + b sin a) = 0,

tFIG. 1-28 FIG. 1-29

34

or

Hence:

CURVES [CH. 1

t cos a + b sin a = a, constant unit vector, along the axis.

cos a = a t.

The curve is therefore of constant slope. We can express this result asfollows:

A necessary and sufficient condition that a curve be of constant slope is thatthe ratio of curvature to torsion be constant. (Theorem of Lancret, 1802;first proof by B. de Saint Venant, Journal Ec. Polyt. 30, 1845, p. 26.)

The equation of a helix can be written in the form (line l is here theZ-axis) :

x = x(s), y = y(s), z = s cos a,which shows that this curve can be considered as a curve on a generalcylinder making a constant angle with the generating lines (cylindricalloxodrome). When K/r = 0 we have a straight line, when K/r = oo, aplane curve.

If we project the helix x(s) on a plane perpendicular to a, the projectionx, has the equation (see Fig. 1-30) :

x,=x -Hence

x; = t - t - a cos a,and its are length is given by

ds; = dx, dx, = sine a ds2.Since

dx,/ds, = csc at - cot a a,its curvature vector is

d2x,/ ds; = K csc2 an,Fia. 1-30

and its curvature K, = K csc2 a. In words:The projection of a helix on a plane perpendicular to its axis has its prin-

cipal normal parallel to the corresponding principal normal of the helix, andits corresponding curvature is K, = K csc2 a.

EXAMPLES. (1) Circular helix. If a helix has constant curvature, thenits projection on a plane perpendicular to its axis is a plane curve of con-stant curvature, hence a circle (Section 1-8). The helix lies on a cylinderof revolution and is therefore a circular helix.

7 -9] HELICES 35

(2) Spherical helices. If a helix lies on a sphere of radius r, then Eq.(8-10) holds, which, together with K = T tan a, gives after elimination of r,

r2 = R2[1 + R'2 tan2 a],or

RdR= +ds cot a,

r

which, integrated for R, and by suitable choice ofthe additive constant in s, gives

R2 + 82 cot2 a = r2.The projection of the helix on a plane perpen-dicular to its axis is therefore a plane curve withthe natural equation

FIG. 1-31

R2 + Si cos2 a = r2 sin4 a.

This type of curve is discussed in Sec. 1-8, and since cos2 a < 1, representsan epicycloid (compare Eq. (8-3)):

A spherical helix projects on a plane perpendicular to its axis in an arc ofan epicycloid.

This projection is a closed curvewhen the ratio a/b of Eq. (8-4) isrational. Using the notation of thisformula and of Eq. (8-3), we obtainthat in this case

B _ acosa = Aa+2b

is rational.From

B _ 4b(a + b = r sine a,a-1-2b

A _ 4b(a + b)a

= r sin a tan a,we find FIG. 1-32.

Only three of the six arcsare drawn.

a = r cos a, b = 2 (1 - cos a) = r sine a/2, (a + 2b = r)for the radius of the fixed and of the rolling circle. When cos a = * wefind as projection an epicycloid of three cusps (Fig. 1-31); each of thethree arcs represents part of a helix on the sphere (Fig. 1-32).

This case has been investigated by W. Blaschke, Monatshefte fur Mathematikand Physik 19, 1908, pp. 188-204; see also his Diferentialgeometrie, p. 41.

36 CURVES [CH. 1

1-10 General solution of the natural equations. We have seen thatthe natural equation of a plane curve can be solved (that is, the cartesiancoordinates can be found) by means of two quadratures. In the case of aspace curve we can try to solve the third order differential equation in a(s)obtained from Eq. (8-6) by eliminating 0 and y. The solution can, how-ever, be reduced to that of a first order differential equation, a so-calledRiccati equation, which is a thoroughly studied type, but not a type whichcan be solved by quadratures only. This method, due to S. Lie and G. Dar-boux, is based on the remark that

a2 + 02 + y2 = 1 (10-1)

can be decomposed as follows:

(a + i13)(a - iQ) + 'Y)(1 - 'Y).Let us now introduce the conjugate imaginary functions w and -z-1:

w+i/3_ 1+y. -1_a-i(_ 1+y (10-2)1-y a-i(3' z 1-y a+i13Then it is possible to express a, 0, y in terms of w and z:

1-wz a_i1+wz y=w+z (10-3)w -z w- z w- z

These expressions are equivalent to the equations of the sphere(Eq. (10-1)) in terms of two parameters w and z. We shall return to thisin Sec. 2-3, Exercise 2.

With the aid of the Eqs. (8-6) we find for w':dw _ a'+ i/3' a+i13 KO - i,a+try wy' r(i7 -(3w)ds 1-y +(1-y)27 = 1-y +1-y=-iKw+ 1-yBecause of Eq. (10-2) we find for S:

1 + ,Y - w2 + 772R_Z2w

The elimination of 0 from these two equations gives an equation from whichy also has disappeared:

ds =-2r-iKw+ 2W2. (10-4)

1-10] GENERAL SOLUTION OF THE NATURAL EQUATIONS 37Performing the same type of elimination for dz/ds, we find that z satis-

fies the same equation as w. This equation has the form

ds = A + Bf + Cf, where A (s) 2, B(s) = -iK, C(s) = 2 (10-5)This is a so-called Riccati equation. It can be shown that its general solu-tion is of the form

c1+f2f Cf3 + f,'where c is an arbitrary constant and the fi are functions of s.

(10-6)

The fundamental properties of a Riccati equation are:(1) When one particular integral is known, the general integral can be obtained

by two quadratures.(2) When two particular integrals are known, the general integral can be found

by one quadrature.(3) When three particular integrals f1, f2, f3 are known, every other integral f

satisfies the equation

f -f2 f3-f2In words: The cross ratio of four particular integrals is constant.* Eq. (10-6)is a direct consequence of Eq. (10-7).

Let fl, f2, f3, f4 be such functions of s that Eq. (10-6) is the generalintegral of Eq. (10-4). In order to find ai, Ai, -ti, i = 1, 2, 3, we needthree integrals wi and three integrals zi, to be characterized by constantsc1, c2, C3 for the wi, and three constants d1, d2, d3 for the zi:

1 - w1z1 pp 1 + wiz1 W1 + Z1 etc.a1= w1-Z1f Ml=i w1-z1r YI =V)1-Z1f

The nine functions ai, $i, yi must satisfy the orthogonality conditionsai + Qi + Y+ = 1, i = 1, 2, 3;

alai + 0i0i + 'Y(Yi = 0, i, j = 1, 2, 3, i P1 j.The first three are automatically satisfied by virtue of Eq. (10-1), and there-fore we have to find the ci, di in such a way that the last three conditions

* Proofs in L. P. Eisenhart, Differential geometry, p. 26, or E. L. Ince, Ordinarydifferential equations (1927), p. 23.

38 CURVES [CH. 1

are also satisfied. One choice of cti, d, is sufficient, since all other choiceswill give curves congruent to the first choice. Now ala2 + 0102 + 'Y1 Y2 = 0can be written as follows:

or

(w1 - w2) (z1 - z2) = - (w1 - z2) (ZI - W2),

2(wlzl + W2Z2) = wlz2 + w2z1 + w1w2 + z1z2,and if we substitute for w1, w2, zl, Z2 their values (see Eq. (10-6)), with theconstants c1, C2, d1, d2 respectively, we obtain the same relation for theconstants:

2(cid, + c2d2) = c1d2 + CA + C1c2 + d1d2.

Similarly:2(C2d2 + C33) = C2d3 + CA + C2C3 + d2d3,2(c3d3 + cidi) = c3d1 + cld3 + C3C1 + d3d1.

Every solution of these three equations in six unknowns c;, d; will give acoordinate expression for the curve. A simple solution is the following:

Cl = 1, c2 = i, c3 = 00 ; d1 = - 1, d2 =-i, d3 = 0.To verify this, substitute into the three equations the values of

c1, c2, d1, d2, which gives CA = - 1 as well as c3d3 = + 1 which is compatiblewith c3 = oo, d3 = 0. The c and d form three pairs of numbers, eachpair of which is harmonic with respect to each other pair. This is a directresult of the properties of the general solution of the Riccati equation.Hence w1 = (ff + f2)/(f3 + f4), etc. We thus obtain for the a;:

1 - wlzl _ (1? - fl) - (f2 - f4)WI - zl 214 - J23(JJJ)

=1-w2,z2= (fl-fl)+(fz-f+)1a2W2 - Z2 L 2(Jlf4 - f213)1 - w3Z3

=f3{4 - fl (10-8)a3 =

W3 - Z3 fIf 4 - { I'2{3'

which results in the following theorem.J J

If the general solution of the Riccati equation

df=-i itds 2 r - ikf + 2 f2

is found in the form (fl, f2, f3, f4 functions of s)M + J2J __Cf3 + J4,

(c arbitrary constant)

EVOLUTES AND INVOLUTES

then the curve given by the equations

x =f sat ds, y = J sae ds, z =Jsa3 ds,

39

where the a; are given by Eq. (10-8), has K(s) and T(s) as curvature and torsion.This reduction to a Riccati equation goes in principle back to Sophus Lie

(1882, Werke III, p. 531) and was fully established by G. Darboux, Lecons I,Ch. 2. We find Eq. (10-8) in G. Scheffers, Anwendung I, p. 298.

EXAMPLES. Plane curve. When r = 0:df/f = -iK ds,('

f =ce', =J Kds,fl = e ", f2 = 0, f3 = 0, f4 = 1,

which lead to the Eqs. (8-2) of the plane curve.Cylindrical helix. In this case the Riccati equation (10-4) can be written

in the formw' _ - ari(1 + 2cw - w2). (c constant)

Two integrals can immediately be found by taking w2 - 2cw + 1 = 0.The general solution of this equation can now be found by means of onequadrature. For the details of this problem we may refer to Eisenhart,Differential geometry, p. 28.

1-11 Evolutes and involutes. The tangents to a space curve x(s)generate a surface. The curves on this surface which intersect the gener-ating tangent lines at right angles form the involutes (German: Evolvente;French: developpantes) of the curve. Their equation is of the form(Fig. 1-33):

y = x + fit. (X a function of s)(11-1)

The vector dy/ds is a tangent vectorto the involute. Hence:

t dy/ds = 0,

1 + = 0'Aconst-s=c-s. FIG. 1-33

40 CURVES

The equation of the involutes is thereforey=x+(c-s)t.

[CH. I

(11-2)For each value of c there is an involute; they can be obtained by unwindinga thread originally stretched along the curve, keeping the thread taut allthe time.

The converse problem is somewhat more complicated: Find the curveswhich admit a given curve C as involute. Such curves are called evolutes ofC (German: Evolute; French: developpees). Their tangents are normal toC(x) and we can therefore write for the equation of the evolute y (Fig. 1-34) :

Hencey=x+ain+a2b.

dLys =

t(1 - a1K) + n ( S1 - rat/d

+ b(dd8a2

+ ra)must have the direction of aln + a2b, thetangent to the evolute:

andK = 1/al, R = a1,

ds' - rat ds2 + rala1 a2

which can be written in the form

a2dR-Rda2a2+R2

This expression can be integrated:

tan-1 R = fr A + const,or

a2 = R [cot (Jr ds + const)].The equation of the evolutes is:y=x+R [n+cot (frd8+const) b]. (11-3)

FIG. 1-34

FIG. 1-35

C

1-111 EVOLUTES AND INVOLUTES 41

This equation shows that a point of the evolute lies on the polar axis of thecorresponding point of the curve, and the angle under which two differentevolutes are seen from the given curve is constant (Fig. 1--35).

For plane curves:y = x + Rn + XRb. (X a constant)

For X = 0 we obtain the plane evolute. The other evolutes lie on thecylinder erected on this plane evolute as base and with generating linesperpendicular to the plane. They are helices on this cylinder.

The theory of space evolutes is due to Monge (1771, published in 1785, thepaper is reprinted in the Applications). Further studies were published byLancret, Menmires presentees et l'Institut, Paris, 1, 1805, and 2, 1811. Lancretdiscussed the "developpoides" of a curve, which are the curves of which thetangents intersect the curve at constant angle not = 90.

The locus of the polar axes is a surface called the polar developable(see Sec. 2-4). On this surface lie the oo' evolutes of the curve and alsothe locus of the osculating spheres. This locus cannot be one of theevolutes, since its tangent has the direction of the binormal b (see Eq.(8-11)), while the tangent to the evolute has the direction Rn + alb.

EXERCISES

1. The perpendicular distance d of a point Q(y) to a line passing through P(x)in the direction of the unit vector u is d = I (y - x) x uI (Fig. 1-36). Using thisformula, show that the tangent has a contact of order one with the curve.

d

IFIG. 1-36 FIG. 1-37

2. The perpendicular distance D of a point Q(y) to a plane passing throughP(x) and perpendicular to the unit vector u is D = I(y - x) uI (Fig. 1-37).Using this formula, show that a plane through the tangent has a contact of orderone, but for the osculating plane which has a contact of order two with the curve.

3. Show (a) that the tangent has a contact of order n with the curve, if x', x",x"', ... x("), but not x("+'), have the direction of the tangent; and (b) that theosculating plane has such a contact, if x', x", x"', ... X("), but not x("+,) lie in theosculating plane.

42 CURVES [CH. 1

4. Show that the distance of a point P(xi) to the plane a x + p = 0 is of theorder of a x, + p, and that of P to the sphere (x - a) (x - a) - r2 = 0 is oforder (x, - a) (x, - a) - r2.

5. Osculating helix. This is the circular helix passing through a point P of thecurve C, having the same tangent, curvature vector, and torsion. Show that itscontact with the curve is of order two. Is it the only circular helix which has acontact of order two with C at P? (T. Olivier, Journal Ecole Potytechn., cah. 24,tome 15, 1835, pp. 61-91, 252-263.) Also show that the axis of the osculating helixis the limiting position of the common perpendicular of two consecutive principalnormals.

6. Starting with the common coordinate equations, find the natural equations of(a) logarithmic spiral:

(b) cycloid:r = cekB.

x = a(0 - sin 0), y = all - cos 0).

(c) circle involute:x = a(cos 0 + 0 sin 0), y = a(sin 0 - 0 cos 0).

(d) catenary:y = (a/2) (erl + e-i).

7. Find from Eq. (8-6), in the case of constant K and r, the third order equationfor a, and by integration obtain the circular helix.

8. Prove that when the twisted cubicx=at, y=bt2, z=t'

satisfies the equation 2b2 = 3a, it is a helix on a cylinder with generating line parallelto the XOZ-plane, making with the X-axis an angle of 45. Determine the equa-tion of the cylinder.

9. The spherical indicatrix of a curve is a circle if and only if the curve is a helix.10. The tangent to the locus C of the centers of the osculating circles of a plane

curve has the direction of the principal normal of the curve; its are length betweentwo of its points is equal to the difference of the radii of curvature of the curve atthese points.

11. Curves of Bertrand. When a curve C, can be brought into a point-to-pointcorrespondence with another curve C so that at corresponding points P1, P the curveshave the same principal normal, then

(a) PIP is constant = a,(b) the tangent to C at P and the tangent to C, at P, make a constant angle a,(c) there exists for C (and similarly for C,) a linear relation between curvature

and torsionK + r cot a = 1/a, ( take r 0, a 0, a 9-6 2) (A)

These curves were first investigated by J. Bertrand (Journal de Mathem. 15,1850, pp. 332-350).

1-11] EVOLUTES AND INVOLUTES 43

12. Show that a curve C for which there exists a linear relation between curva-ture and torsion:

PK + Qr = R, (P, Q, R constants Fx- 0)admits a Bertrand mate, that is, a curve of which the points can be brought into aone-to-one correspondence with the points of C such that at corresponding pointsthe curves have the same principal normal.

13. Special cases of Bertrand curves. Show:(a) When r = 0 every curve has an infinity of Bertrand mates, a = 0.(b) When r 54 0, a = a/2, we have curves of constant curvature. Each of the

curves C and C, is the locus of the centers of curvature of the other curve.(c) When K and r are both constant, hence in the case of a circular helix, there

are an infinite number of Bertrand mates, all circular helices.14. Show that the equation of a Bertrand curve (that is, a curve for which (A),

Exercise 11, holds) can be written in the form

x= a fu dv + a cot a fu x du, (a, a constants)Here u = u(v) is an arbitrary curve on the unit sphere referred to its arc length(hence u u = 1, u' u' = 1, u' = du/dv). (Darboux, Lecons I, pp. 42-45.)Show that the first term alone on the right-hand side gives curves of constant curva-ture, the second term alone curves of constant torsion.

15. Mannheim's theorem. If P and P, are corresponding points on two Bertrandmates, and C and C, their centers of curvature, then the cross ratio (CC,, PP,) _sec' a = const.

16. Investigate the pairs of curves C, and C which can be brought into a point-to-point correspondence such that at corresponding points (a) the tangents are thesame, (b) the binormals are the same.

See E. Salkowski, Math. Annalen 66, 1909, pp. 517-557, A. Voss, Sitzung8ber.Akad. Miinchen, 1909, 106 pp., where also other pairs of corresponding curves arediscussed.

17. Show that when the curve x = x(s) has constant torsion r, the curve

has constant curvature r.

y=-Tn+ Jb ds18. Show that if in Sec. 1-10 we split a2 + 02 + y2 = 1 into (a + iy) (a - iy) =

(1 + #)(1 - 0) and follow the method indicated by Eq. (10-2), we are led to theRiccati equation

LJ' = -$(K - 2r) + (K + ir)w2.19. Loxodromes. These can be defined as curves which intersect a pencil of

planes at constant angle a. Show that their equation can be brought into the formx = r cos 0, y = r sin 0, z = f (r), where r = Vx2 + y2

and

0=tans f 1+(f')2ar.J r

44 CURVES [CH. 120. The tangents to a helix intersect a plane perpendicular to its projecting

cylinder in the points of the involute of the base of the cylinder.21. Find the involutes of a helix.22. Show that the helices on a cone of revolution project on a plane perpendicu-

lar to their axes (the base) as logarithmic spirals and then show that the intrinsicequations of these conical helices are

R = as, T = bs, (a, b constants)23. Show that the helices on a paraboloid of revolution project on a plane per-

pendicular to the axis as circle involutes.24. A necessary and sufficient condition that a curve be a helix is that

(x "xrrrar1)= -K5 ) = 0.

Ws- (K25. Differential equation of space curves. If, instead of using the method of the

Riccati equation, we attempt to obtain the x(s) directly as the solution of theFrenet equation for given K and r, we obtain the equation (K, TO 0)

I q 2 'a(iv) -

2K + T arrr + K2 + T2 _ KK - 2(K') + KY) Z + K2 (K - r ) Sr = 0.K T K2 KT K T

26. The parabolas y = x2 and y = N"tr can be obtained from each other by arotation of 90. They must therefore satisfy the same natural equation. Obtainthis equation in the form

1ds/2=9['482-11.

27. Verify by using Eq. (8-12) that the curve of Exercise 9, Section 1-6:x = a sin2 u, y = a sin u cos u, z = a cos u,

lies on a sphere.

1-12 Imaginary curves. We have so far admitted only real curves, de-fined by real functions of a real variable. When we admit complexanalytic functions x; of a complex variable u, then we obtain structuresof 002 points, called imaginary curves. The formulas for the are length,tangent, osculating plane, curvature, and torsion retain a formal meaningfor those curves, but now they serve as definition of these concepts. Cer-tain theorems require modification, notably those based on the assumptionthat ds2 > 0. We have to admit, in particular, curves for whicl` ds2 = 0,the isotropic curves. And if we define planes as structures for which alinear relation exists between the x;, or - what amounts to the same - asplane curves, curves for which (x a a) = 0, then we have to admit planessuch as the plane with equation x2 = ixi, for which ds2 can no longer bewritten as the sum of the squares of two differentials, but as the square of

1-12] IMAGINARY CURVES 45

a single differential such as ds2 = dxi in the case of x2 = ix1. We thereforedistinguish between regular planes, of which the equation by proper choiceof the coordinates can be reduced to x3 = 0, and isotropic planes, of whichthe equation can be reduced to x2 = ix1. By planes we mean regularplanes. There are no planes for which ds2 = 0. (Such planes do exist infour-space; e.g., those for which x1 = ix2, x3 = 2x4.)

Isotropic curves in the regular plane x3 = 0 are defined by

ds2 = dxi + dx2 = 0; dx2 = i dx1, (12-1)which equation gives, by integration:

X2 - b = i(xi - a). (12-2)In an isotropic plane we obtain by integration of ds2 = dxi = 0 that x3 = c,x2 = ix1. Hence:

Plane isotropic curves are straight lines. Through every point of a regularplane pass two isotropic lines; through every point of an isotropic plane passesone isotropic line. The isotropic lines in a regular plane form two sets ofparallel lines, those in an isotropic plane form one set of parallel lines.

Isotropic lines were introduced by V. Poncelet in his Traite des propriltesprooectives des figures (1822). They are also called minimal curves or null curves.Isotropic curves in space x = x(u) satisfy the differential equation

dx- dx = dxl + dx2 + dxi = 0, (12-3)or

a a = 0, x = dx/du. (12-4)One solution of this differential equation consists of the isotropic straight

linesX = x + ua, x, a constant vectors, a a = 0, (12-5)

for which, as we see by differentiation, dX dX = a a du2 = 0. Theselines generate, at each point P(x), a quadratic cone

(X - x) (X - x) = 0, (12-6)the isotropic cone with vertex P. The tangent plane to this cone along thegenerator (12-5) has the equation

(X - (12-7)This is easily verified if we remember that the tangent plane to the cone

x2 + y2 + z2 = 0 at (x,, y1, z1) is xx1 + yy1 + zz1 = 0, and that this plane remainsthe same when we multiply x1, y1, z, by a factor X, so that this plane is tangentalong the generator x : y : z = x1 : y1 : z1 = a1 : as : as.

46 CURVES [CH. 1

By selecting x = 0 and taking the XOY-plane through a, the equation ofplane (12-7) becomes x1 = ix2, so that (12-7) is an isotropic plane.Eq. (12-7) is, at the same time, the most general form of the equation of anisotropic plane, since a a = 0 for the planes x1 = ix2 and therefore alsozero for all positions of a rectangular cartesian coordinate system.

Substitution of the X of Eq. (12-5) into Eq. (12-7) shows that isotropicline (12-5) lies in the plane (12-7). Hence:

The tangent planes to the isotropic cones are isotropic planes, tangent alongisotropic lines, and also, although it seems strange at first:

The normal to an isotropic plane at a point in this plane lies in this planeand is tangent to the isotropic cone at that point.

Isotropic straight lines do not form the only solution of Eq. (12-4).We shall now derive the general solution by a method reminiscent of thatused to pass from Eq. (10-1) to Eq. (10-2). For this purpose we writeEq. (12-3) in the form

dx1 + i dx2 dx3= u.dx3 dx1 - i dx2

This leads to the equations

dx1 dx2 _ dxj dx2 1dx3+Z dx3-u, dx3-idx3=-u

Solving these two equations for dxi/dx3 and dx2/dx3, we obtain the solu-tion of Eq. (12-4) in the form

dx1 _ dx2 _ dx3= F(u) du.

u2 - 1 i(U2 + 1) 2uWriting f"'(u) for F(u), we then obtain by partial integration the equation of

the isotropic curves in the form:

x1 = (u2 - 1)f" - 2uf' + 2f,x2 = [(u2 + 1)f" - 2uf + 2fli,x3 = 2uf" - 2f, (12-8)

where f(u) is an arbitrary analytic function of a parameter u. This repre-sentation contains, as we see, no integration. It does not give the(straight) isotropic lines.

The tangent line to an isotropic curve at P(x) is given byy = x + aa,

the osculating plane by(12-9)

y = x + Xi + %, or (y - x, x, x) = 0. (12-10)

1-13] OVALS 47

From Eq. (12-4) followsx x = 0. (12-11)

Since(a X a) . (g X a) = (g j[) (:k g) - (i g)2, (12-12)

we see that(i X :R) (s x it) = 0. (12-13)

Hence, comparing with Eq. (12-7), we find that the osculating plane of anisotropic curve (not a straight line) is tangent to the isotropic cone.

However, all curves whose osculating planes are tangent to the isotropiccone are not all isotropic curves. Such curves are characterized byEq. (12-13). In this c